Set-up

Currently, we work with 19 years of daily maximum temperature data over summer months (June, July, August). (The issue with year 20 remains unresolved, last year was removed.)

Add all sites together.

Follow-up from meeting with Simon: exploring \(\alpha\) against distance

Recall the plot of marginal estimates of \(\alpha\) parameter against distance[m] for the three conditioning sites. This is done using the sequential method (see below).

We look at Birmingham (left) and London (right).

Link both back to spatial locations to explore any potential patterns.

Birmingham and London show a similar pattern, which suggest higher dependence decay with distance (black) in the south of the mainland UK and in the vicinity of the conditioning sites. The dividing line could be moved to explore this result further.

Sequential parameter estimation

  1. Fix \(\beta=0\), estimate \(\hat{\alpha}\).
  2. Fix \(\alpha=\hat{\alpha}\), estimate \(\hat{\beta}\).
  3. Fix \(\alpha=\hat{\alpha}\) and \(\beta=\hat{\beta}\), estimate \(\hat{\mu}\) and \(\hat{\sigma}\).
  4. Calculate observed residuals, estimate \(\hat{\mu}_{AGG}\), \(\hat{\sigma}_{AGG}\), \(\hat{\delta}_l\) and \(\hat{\delta}_u\).
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Now, the same approach with alpha values on the fitted exponential curve with distance, so we use estimates of \(\hat{\alpha}\) to fit an exponential curve to as a function of distance of site \(i\) from the conditioning site \(j\).

  1. Fix \(\alpha=\alpha \left( d_{ij} \right) = \exp \left\{ - \phi d_{ij} \right\}\), estimate \(\hat{\beta}\).
  2. Fix \(\alpha=\alpha \left( d_{ij} \right)\) and \(\beta=\hat{\beta}\), estimate \(\hat{\mu}\) and \(\hat{\sigma}\).
  3. Calculate observed residuals, estimate \(\hat{\mu}_{AGG}\), \(\hat{\sigma}_{AGG}\), \(\hat{\delta}_l\) and \(\hat{\delta}_u\).

We can observe that imposing a parametric form on \(\hat{\alpha}\) that only changes with distance leads to some unexpected values of \(\hat{\beta}\) close to \(1\). For the AGG distribution estimates, \(\hat{\mu}_{AGG}\) is the mode of the distribution whereas \(\hat{\mu}\) is the median as well as the mode for the symmetric Gaussian distribution. Therefore, lower values for \(\hat{\mu}_{AGG}\) suggest heavier tails for the upper tail of the distribution. This is also shown by a positive difference of \(\hat{\sigma}_u-\hat{\sigma}_l\).

New iterative approach

Try a parametric alpha decaying exponentially with distance.

First, we plot the iterations of parameter estimates to check convergence.

Repeat also for Glasgow.

and London.

Plot also on a map.

Conditioning on any random site

Currently, the analysis is conditioning on Birmingham, Glasgow and London, ordered as first 3 columns of the temperature data for an easy link.

One way would be to calculate distance from each of the site and then filter distance for the site needed, which will be faster as this distance matrix will be already saved (twice as \(d_{ij}\) and \(d_{ji}\) are identical ).

We find the index of the grid sites closest to Inverness, Lancaster and Newcastle to get a better idea of potential parametric forms of the parameters.

Sequential parameter estimation: other conditioning sites

We repeat the parameter estimation method described above for three other conditioning sites: Inverness, Lancaster and Newcastle.

Examine similar plots of parameter estimate iterations for a randomly picked site \(j\) (set to \(j=100\)).

Inverness:

Lancaster:

Newcastle:

Similar to the previous section, we map the parameter estimates of the final iteration.

Sequential parameter estimation: \(3\) new sites

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Now fit exponential to \(\alpha\) estimates and find MLE for the other parameters using the sequential approach.

New sites: Cromer, Hull and Lowestoft

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Explore time dependence

Run for all the 12 sites.

Map and save for all sites.

Save max over the time period and which day it is.

Explore high values of \(\beta\)

##           a            b res cond_site tau
## 1 0.4385825 6.214449e-01 172     Truro  -3
## 2 0.3016202 3.258110e-01 172     Truro  -2
## 3 0.2872649 1.639558e-07 172     Truro  -1
## 4 0.3315165 3.125944e-07 172     Truro   0
## 5 0.2793784 3.307378e-01 172     Truro   1
## 6 0.2874797 3.437228e-01 172     Truro   2
## 7 0.3471748 3.483853e-01 172     Truro   3

Scatterplot of \((Y_{172},Y_{Truro}| Y_{Truro}>v\).

##   lik     lika     likb      lik2         a         b mu     mu_agg sig sig_agg
## 1  NA 282.5537 271.2819 -161.4081 0.4385825 0.6214449  0 -0.7494323   1      NA
##        sigl      sigu delta   deltal   deltau given res
## 1 0.3804989 0.8750895    NA 1.271873 1.216345     1   2

Explore how a scatterplot varies for different \(\tau\).

Explore quantiles of the data (on the Laplace scale).